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Technical Report on FFT Analyzer 12

7. FFT and Time Windows

7.1 Rectangular Window

(Equation 7-2)

The rectangular pulse p(t) defined here is referred to as a rectangular window (square wave window function). This is simply the same as the original data cut off at a time length T. This Fourier transform is as stated in the previous section;

Equation 7-3

When represented in a diagram, P(ω) appears as shown in Figure 7-5. The upper part of the diagram represents the shape of the time window, while the lower part shows the logarithmic absolute value of the spectrum, with the units on the frequency axis being the inverse of the time window length.

  • Figure 7-5

Note that Figure 7-5 is normalized to display as 1 when f 0 is used. This allows for direct reading of the amplitude at f 0. Please be aware that all subsequent illustrations are displayed in a normalized manner.

Adaptation to f (t) ＝ cos ω 0 t

Consider f(t) as cos ω 0 t. The cos 2π f 0 t・ w  (t) will be a waveform where the cos waveform of frequency f 0 has its amplitude changed by a factor of w  (t) (amplitude modulated).

The Fourier transform of the waveform cut out by the square wave window is;

Equation 7-4

than

Equation 7-5

Adaptation to f (t) = sin ω 0 t

Similarly, considering a sine wave instead of a cosine wave:

Equation 7-6

The first and second terms of equations 7-5 and 7-6 represent waveforms that shift the center of P (ω) to ±f 0, and equation 7-5 takes their sum, while equation 7-6 takes their difference. The waveform of equation 7-6 when f 0 = 3 /T is shown in Figure 7-6. Now, the DFT value in Figure 7-6 takes points of the harmonics at frequency 1/T, so looking at the value of that point:

When f 0 (n = 0): W (f 0) = 1

When f 0 ± n /T (n = 1, 2, .....) : W (f) = 0

It well represents that sin ω 0 t has a frequency of f 0.

This is because it is cut to match the period of the sine waveform as shown in the previous figure 7-1, making the start and end points continuous.

  • Figure 7-6

Next, considering the case where it is cut off without matching the period f 1 = 3.5 / T, as shown by the vertical line in Figure 7-7;

f 1 (n = 0): ≠ 0

f 1 ± n / T (n = 1, 2, 3, ..... ) when: ≠ 0

It has a value of f 1 ± n / T. Unlike Figure 7-5, where DFT does not have a point for f 1, it will have a harmonic spectrum of f 0. This is an error caused by the discontinuity of the starting and ending points of the cut, known as leakage error. When analyzing signals that have a spectrum of f n in addition to f 0, if the component of f n is too small, it will be buried under the leakage error of f 0. (Please refer to the rectangular window in Figure 8-3.)

  • Figure 7-7

Figure 7-8 shows the overlay of Figures 7-6 and 7-7. As can be seen from this figure, the Y-axis value C at the point 3.5 / T when f 0 = 3 / T and the value B at 3 / T when f 1 = 3.5 / T are the same. Considering that the spectra other than B in Figure 7-7 will be similar, we can think of the waveform at f 1 = 3.5 / T as having passed through the bandpass filter P (ω) (which is a shifted version of Figure 7-5 at f 0).

  • Figure 7-8

From the above, as a concept of the FFT analyzer, the DFT that seeks the spectrum of the n-th harmonic with f 0 as the fundamental frequency has the shape of a bandpass filter W(f), and can be considered as being replaced by a number of bandpass filters arranged with their center frequency shifted by f 0 while maintaining n. This is illustrated in Figure 7-9.

  • Figure 7-9

A rectangular window is a reasonable window for a signal like a shock waveform, where the start and end points are zero, but for continuous signals, it does not provide good frequency separation due to leakage errors.

As we have seen so far, the changes in the power spectrum due to window functions depend on the characteristics of the Fourier transform of the window function, so it suffices to investigate the properties of W(f).

This average power is the same in both the frequency domain and the time domain according to Parseval's theorem.

(Equation 7-7)

When normalized and expressed in logarithmic form, it becomes 0dB, and the reciprocal of this corresponds to the formula for variance b. This represents the bandwidth under the assumption of equal power and is one of the parameters that characterize the window.